Need help on how to solve system of equations 2x - y + z = -7, x - 3y + 4z = -19, and -x + 4y - 3z = 18.
Not one of these answer help at all
To solve a system of equations, we can use the method of elimination or substitution. I will explain both methods, and you can choose the one you find more convenient.
Method 1: Elimination
Step 1: Multiply each equation by a suitable factor to make the coefficients of one of the variables the same in two equations. In this case, let's make the coefficients of x in the first and second equations the same.
- Multiply the first equation by 1.
- Multiply the second equation by 2.
Step 2: Now, add the two equations together to eliminate the x variable.
(2x - y + z) + 2(x - 3y + 4z) = -7 + 2(-19)
Simplify the equation:
2x - y + z + 2x - 6y + 8z = -7 - 38
Combine like terms:
4x - 7y + 9z = -45
Step 3: Repeat steps 1 and 2 to eliminate another variable. Let's eliminate the x variable again.
- Multiply the first equation by 1.
- Multiply the third equation by 1.
Step 4: Add the two equations together to eliminate the x variable.
(2x - y + z) + (-x + 4y - 3z) = -7 + 18
Simplify the equation:
2x - y + z - x + 4y - 3z = -7 + 18
Combine like terms:
x + 3y - 2z = 11
Step 5: Now, we have a system of two equations with two variables:
4x - 7y + 9z = -45 (from step 2)
x + 3y - 2z = 11 (from step 4)
You can solve this system of equations using the elimination method, substitution method, or any other method you prefer. However, these two equations are a bit more complex to solve by hand. If you have access to a graphing calculator or software, you can input the equations and find the point of intersection, which will give you the solution.
add the 2nd and 3rd:
y + z = -1 ... #4
Twice the 3rd added to the first:
7y -5z = 29 .... #5
5 times #4 ---> 5y+5z = -5
#5 ---------> 7y - 5z = 29
add them:
12y = 24
y = 2
in #4
2 + z = -1
z = -3
in #1
2x - 2 - 3 = -7
x = -1
x=-1, y=2 , z=-3